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6.2 EXAMPLE standard deviation that is less than bonds. Using Equation 6.3 again, we find that the portfolio
Benefits from variance is
Diversification 12)2 25)2
(0.75 (0.25 2(0.75 12)(0.25 25) 0 120
and, accordingly, the portfolio standard deviation is 120 10.96%, which is less than the
standard deviation of either bonds or stocks alone. Taking on a more volatile asset (stocks) ac-
tually reduces portfolio risk! Such is the power of diversification.


We can find investment proportions that will reduce portfolio risk even further. The risk-
minimizing proportions will be 81.27% in bonds and 18.73% in stocks.1 With these propor-
tions, the portfolio standard deviation will be 10.82%, and the portfolio™s expected return will
be 6.75%.
Is this portfolio preferable to the one with 25% in the stock fund? That depends on investor
investment
preferences, because the portfolio with the lower variance also has a lower expected return.
opportunity set
What the analyst can and must do, however, is to show investors the entire investment
Set of available
opportunity set as we do in Figure 6.3. This is the set of all attainable combinations of risk
portfolio risk-return
and return offered by portfolios formed using the available assets in differing proportions.
combinations.
Points on the investment opportunity set of Figure 6.3 can be found by varying the invest-
ment proportions and computing the resulting expected returns and standard deviations from
Equations 6.2 and 6.3. We can feed the input data and the two equations into a personal com-
puter and let it draw the graph. With the aid of the computer, we can easily find the portfolio
composition corresponding to any point on the opportunity set. Spreadsheet 6.5 shows the in-
vestment proportions and the mean and standard deviation for a few portfolios.


The Mean-Variance Criterion
Investors desire portfolios that lie to the “northwest” in Figure 6.3. These are portfolios with
high expected returns (toward the “north” of the figure) and low volatility (to the “west”).
These preferences mean that we can compare portfolios using a mean-variance criterion in the
following way. Portfolio A is said to dominate portfolio B if all investors prefer A over B. This
will be the case if it has higher mean return and lower variance:
E(rA) E(rB) and A B


1
With a zero correlation coefficient, the variance-minimizing proportion in the bond fund is given by the expression:
2 2 2
S /( B S).
Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition




179
6 Efficient Diversification




F I G U R E 6.3
12
Investment
Expected return (%)
11 opportunity set for
Stocks
10 bond and stock funds
9 Portfolio Z
8
7 The minimum
variance portfolio
6
Bonds
5
4
6 11 16 21 26 31 36
Standard deviation (%)




S P R E A D S H E E T 6.5
Investment opportunity set for bond and stock funds


A B C D E

Data
1
σS σB ρ SB
E(rS) E(rB)
2

10% 6% 25% 12% 0%
3

Portfolio Weights Expected Return
4
wS wB = 1 - wS E(rP)=Col A A3 Col B B3 Std. Deviation*
5

0 1 6.00% 12.00%
6

0.1 0.9 6.40% 11.09%
7

0.1873 0.8127 6.75% 10.8183%
8

0.2 0.8 6.80% 10.8240%
9

0.3 0.7 7.20% 11.26%
10

0.4 0.6 7.60% 12.32%
11

0.5 0.5 8.00% 13.87%
12

0.6 0.4 8.40% 15.75%
13

0.7 0.3 8.80% 17.87%
14

0.8 0.2 9.20% 20.14%
15

0.9 0.1 9.60% 22.53%
16
1 0 10.00% 25.00%
17

Note: The minimum variance portfolio weight in stocks is
18
wS=(σB^2-σBσSρ)/(σS^2+σB^2-2*σBσSρ)=.1873
19

* The formula for portfolio standard deviation is:
20
σP=[(Col A*C3)^2+(Col B*D3)^2+2*Col A*C3*Col B*D3*E3]^.5
21




Graphically, if the expected return and standard deviation combination of each portfolio
were plotted in Figure 6.3, portfolio A would lie to the northwest of B. Given a choice between
portfolios A and B, all investors would choose A. For example, the stock fund in Figure 6.3
dominates portfolio Z; the stock fund has higher expected return and lower volatility.
Portfolios that lie below the minimum-variance portfolio in the figure can therefore be re-
jected out of hand as inefficient. Any portfolio on the downward sloping portion of the curve
is “dominated” by the portfolio that lies directly above it on the upward sloping portion of the
curve since that portfolio has higher expected return and equal standard deviation. The best
choice among the portfolios on the upward sloping portion of the curve is not as obvious,
Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition




180 Part TWO Portfolio Theory


because in this region higher expected return is accompanied by higher risk. The best choice
will depend on the investor™s willingness to trade off risk against expected return.
So far we have assumed a correlation of zero between stock and bond returns. We know
that low correlations aid diversification and that a higher correlation coefficient between
stocks and bonds results in a reduced effect of diversification. What are the implications of
perfect positive correlation between bonds and stocks?
Assuming the correlation coefficient is 1.0 simplifies Equation 6.3 for portfolio variance.
Looking at it again, you will see that substitution of BS 1 in Equation 6.3 means we can
“complete the square” of the quantities wB B and wS S to obtain
2 2
(wB wS S)
P B

and, therefore,
wB wS
P B S

The portfolio standard deviation is a weighted average of the component security standard
deviations only in the special case of perfect positive correlation. In this circumstance, there
are no gains to be had from diversification. Whatever the proportions of stocks and bonds,
both the portfolio mean and the standard deviation are simple weighted averages. Figure 6.4
shows the opportunity set with perfect positive correlation”a straight line through the com-
ponent securities. No portfolio can be discarded as inefficient in this case, and the choice
among portfolios depends only on risk preference. Diversification in the case of perfect posi-
tive correlation is not effective.
Perfect positive correlation is the only case in which there is no benefit from diversifica-
tion. Whenever 1, the portfolio standard deviation is less than the weighted average of the
standard deviations of the component securities. Therefore, there are benefits to diversifica-
tion whenever asset returns are less than perfectly correlated.
Our analysis has ranged from very attractive diversification benefits ( BS 0) to no bene-
fits at all ( BS 1.0). For BS within this range, the benefits will be somewhere in between. As
Figure 6.4 illustrates, BS 0.5 is a lot better for diversification than perfect positive correla-
tion and quite a bit worse than zero correlation.



F I G U R E 6.4 12
Investment opportunity
sets for bonds and 11
stocks with various
correlation coefficients Stocks
10
ρ
Expected return (%)




1
ρ 0
9

ρ 1
8 ρ 0.5
ρ 0.2
7

Bonds
6

5

4
0 5 10 15 20 25 30 35 40
Standard deviation (%)
Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition




181
6 Efficient Diversification


A realistic correlation coefficient between stocks and bonds based on historical experience
is actually around 0.20. The expected returns and standard deviations that we have so far
assumed also reflect historical experience, which is why we include a graph for BS 0.2 in
Figure 6.4. Spreadsheet 6.6 enumerates some of the points on the various opportunity sets in
Figure 6.4.
Negative correlation between a pair of assets is also possible. Where negative correlation
is present, there will be even greater diversification benefits. Again, let us start with an ex-
treme. With perfect negative correlation, we substitute BS 1.0 in Equation 6.3 and sim-
plify it in the same way as with positive perfect correlation. Here, too, we can complete the
square, this time, however, with different results
2 2
(wB wS S)
P B

and, therefore,
ABS[wB wS S] (6.4)
P B

The right-hand side of Equation 6.4 denotes the absolute value of wB B wS S. The solution
involves the absolute value because standard deviation is never negative.
With perfect negative correlation, the benefits from diversification stretch to the limit.
Equation 6.4 points to the proportions that will reduce the portfolio standard deviation all the




S P R E A D S H E E T 6.6
Investment opportunity set for bonds and stocks with various correlation coefficients

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